Homework Statement
I was trying to prove something and I ended up in a situation similar to,
(limit t\rightarrow0)(limit s\rightarrow0) f(x+s,y+t)
=(limit s\rightarrow0)(limit t\rightarrow0)f(x+s,y+t)
My question is when does this equality hold. I can't find it anywhere...
Homework Statement
I am curious if all modules contain 0.
Homework Equations
A left R-module M over the ring R consists of an abelian group (M, +) and an operation R × M → M such that certain properties hold...
The Attempt at a Solution
The definition of a module says that it is an...
Homework Statement
I am curious,
if I,J, and M are ideals of the commutative ring R, and M/I\subseteqJ/I, then M\subseteqJHomework Equations
M/I = { m+I : m is in M}
J/I = { j+I : j is in J}
I\subseteqR is an ideal if
1.) if a and b are in I then a+b is in I
2.) if r is in R and a is in I...
Ok. So I get 32=9=0 mod 9. So 32 is an element of 9z. By definition 3 is an element of 9z. So I get that it holds for this case. That's what I was supposed to get right?
Homework Statement
In the process of trying to prove something else I found it would be helpful if rn\inI, where I is an ideal, n\inN, and r\inR and R is a ring, then r is in I.
Homework Equations
I is an ideal if a\inI and b\inI then a+b\inI, a\inI and r\inR then ar\inI, and I is not the empty...
Homework Statement
My question is just on the definition of L∞.
Is L∞=Lp where p=∞, i.e.,
is a measurable function in L∞ if ∫Alf(x)l∞<∞?
Homework Equations
*L∞: The space of all bounded measurable functions on [0,1] (bounded except for possibly on a set of measure zero)
*A measurable...
Homework Statement
My question is would I be allowed to say,
if lf+-\phil<ε/(2\mu(E)
then ∫E lf+-\phil<ε/2
Homework Equations
E is the set in which we are integrating over.
\mu is the measure
\varphi is a simple function
f+ is the non-negative part of the function f.
The Attempt...
Homework Statement
I was wondering if we Let E be some set such that f-1(E) is measurable then so is f-1(E)c.Homework Equations
If the set A is measurable then so is its compliment.
The Attempt at a Solution
I think the statement is true because f-1(E) is just a set and thus its compliment...
I have a simple question about differentials. I have been taught two ways to find the differential and my questions is in what situations do I use each one?
simply speaking these are the 2 ways
1.) just take the partials of each component function and throw them in a matrix
2.) Let f be the...
Homework Statement
Does anyone know the process for finding the differential of of f:S→S' where S,S' are surfaces.
My textbook explains how to do this when f is a vector valued function but in the problem that I am working on I have something like f(x,y)=(g(x),h(x),j(y)) rather than something...
Homework Statement
I have this theorem which I am having trouble understanding due to notation.
Theorem: Define S,S'\subseteqℝ3 to be surfaces. Let f:S→S' be a smooth map. f is a local isometry if for all p in S, and all w1,w2 in TpS,
<w1,w2>=<dfp(w1),dfp(w2)>.
The thing I don't...